Assignment 10 solution

  1. Text, problem 6.18.
  2. Text, problem 6.19.
  3. Text, problem 6.27.
  4. You want to drive a certain distance D in a certain time, so the distance function D(t) you have driven after a time t has fixed endpoints. Your gas consumption depends on your car's speed v = dD/dt; assume it is proportional to (dD/dt). Your rate of spending money on gas would also be proportional to this, except that the price of gasoline is increasing. Assume the increase is linear, so your time rate of spending money is proportional to t(dD/dt). (If you (naively) believe gas prices will not increase too fast, take your starting time t1 considerably larger than the trip's length, t2 t1). How should your speed depend on time, and what function should D(t) be, in order to spend the least amount of money on gas for the given trip?
  5. A normally straight metal bar is clamped at the ends so it is somewhat bent. Let y(x) describe its shape. Because of the stiffness of the metal, work has to be done to bend it, which is stored as energy in the bent metal. A measure of the bending is dy/dx, so it is reasonable to assume that the potential energy/length of bar is proportional to (dy/dx)2. Write an integral I for the total energy in the bar and minimize this integral to find its equilibrium shape.
    Here you cannot use the Euler-Lagrange equation directly because the integrand involves a second derivative. Instead, find δI as usual by varying the integrand (analogous to Eq. 6.10). Next, to get the equation analogous to 6.12 you will haved to integrate by parts twice. Thus find the variational equation analogous to 6.13 and write the general solution. What is the shape y(x) of a bar that has been bent into a "U" shape by clamping it with a slope -1 at x = -1 y = 0, and slope +1 at x = +1, y = 0?